Constrained Maximization

1. Find the bundle (X1, 2) that maximizes the function f(x1, 2) = X1X2 subject to 1 + 2 = 2. 2. Find the bundle (1, X2) that maximizes the function f(X1, X2) = 3x125 subject to Pill + P2X2 = I, where p1 = 5, p2 = 25, and I = 100. 3. Consider a consumer whose utility function is U(x1, 22) = 2v122 with an income of $Y, where the price of good 1 is p1 and the price of good 2 is p2. Find the utility maximizing consumption bundle. 4. A consumer maximizes the utility function a In(x1) + BIn(x2) + yIn(x3) subject to the budget constaint 11 + P2x2 + P3X3 = I. Solve for the optimal consumption of x1, 2, and X3. 5. Solve the following constrained optimization problem: max VIn(x1) + In(202) + 23 21 , 2 , 203 subject to 5x1 + 5x2 + 5x3 = 15 6. Derive the FOCs for the following problem (you don't have to solve them): max x1 + 2x2 *1,22 subject to x1 + 2 = 327. Suppose we have a renery that must ship nished goods to some storage tanks. Suppose further that there are two pipelines, A and B , to do the shipping. The cost of shipping :1: units on A is 031102; the cost of shipping 3; units on B is 6312, where a > 0 and b > 0 are given. How can we ship Q units While minimizing cost? What happens to the cost if Q increases by T%? 8. Could the vector (:31, 3:2, 3:3, 334) = (1,9, 5,4) be a solution to the problem: max ln(m1) + 2510233 + 1134 $1?$2!m35m4 subject to 4:51:32 + 33% + 43:4 = c